Statics and Influence Functions From a Modern Perspective

Statics and Influence Functions From a Modern Perspective

Friedel Hartmann Peter Jahn Statics and Influence Functions From a Modern Perspective 123

Friedel Hartmann Department of Civil Engineering University of Kassel Kassel Germany Peter Jahn Department of Civil Engineering University of Kassel Kassel Germany ISBN 978-3-319-51221-1 ISBN 978-3-319-51222-8 (ebook) DOI 10.1007/978-3-319-51222-8 Library of Congress Control Number: 2016963186 Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

The original version of the book was revised: For detailed information please see erratum. The erratum to this book is available at 10.1007/978-3-319-51222-8_9

Preface The new is the old and the old is mightier than ever before. The subject of this book are influence functions and the role they play in finite element analysis of structures. Influence functions are a classical tool of structural analysis and dearly loved by old school engineers since with some clever sketches if need be on a beer mat it is easy to understand the behavior of a structure or to find the weak spots in a design. Unfortunately, with the advent of finite element programs though, the application of influence functions has faded into the background. When in doubt, one rather studies the variants in a design with the computer than to strive for a deeper understanding which the study of influence functions can provide so easily and so well. But new results have rekindled the interest in influence functions because we know today that in linear analysis, finite elements compute everything as we are tempted to say with influence functions. This equals a loop backward. Classical hand methods seemed outdated and old-fashioned, but in finite elements, they have risen like Phoenix from the ashes. FE-analysis is more classical than we ever imagined. In the old days, the subject of influence functions mostly focused on the analysis of frames with the Müller Breslau principle, but in FE-analysis, the concept of influence functions has a much broader and wider scope. The key word is functionals. The deflection at the midspan of a beam, the bending moment at a fixed edge, and the force in a pier all these are functionals. Anything you can calculate with finite elements is considered a functional. And to each linear functional belongs a Green s function, an influence function. Influence functions, too, are displacements, they are the reaction of a structure to special point loads, to Dirac deltas, but normally, a mesh is not that flexible enough to generate the exact intricate shape and exact peaks of these influence functions. vii

viii Preface This is why FE-results are not exact. FE-programs operate with approximate, with substitute influence functions. The influence functions are the real shape functions, the physical shape functions. A good approximation of these functions is the key to good FE-results. So in structural analysis and we dare say all of linear computational mechanics influence functions play a dominant role. This is why we have written this book. It is not a book for a first course in structural mechanics, the reader should be acquainted with the basic principles, and the reader should have seen influence functions in action. We treat the topic also, as it seems, with a rather sharp pencil, but this is more or less self-defense because in the advent of time many things in structural analysis have so much settled in that it is hard to discover the mathematics behind the formulas all too often the ubiquitous dw e = dw i is considered sufficient proof. Kassel, Germany October 2016 Friedel Hartmann Peter Jahn

Contents 1 Basics... 1 1.1 Introduction... 1 1.1.1 Principle of Virtual Displacements... 2 1.1.2 Betti s Theorem.... 3 1.1.3 Influence Functions... 4 1.1.4 Identities... 6 1.2 Green s Identities... 7 1.2.1 Longitudinal Displacement uðxþ of a Bar... 7 1.2.2 Shear Deformation w s ðxþ of a Beam... 8 1.2.3 Deflection w of a Rope... 9 1.2.4 Deflection w of a Beam.... 9 1.2.5 Deflection w of a Beam, Second-Order Theory.... 10 1.2.6 Beam on Elastic Support... 10 1.2.7 Tensile Chord Bridge... 10 1.3 Variational Principles of Structural Analysis.... 11 1.4 Zero Sums... 12 1.5 Examples... 14 1.5.1 The Principle of Virtual Displacements... 14 1.5.2 Conservation of Energy.... 17 1.5.3 The Principle of Virtual Forces... 17 1.6 Frames... 19 1.7 Single Forces and Moments... 21 1.8 Support Settlements... 22 1.9 Springs... 26 1.10 Temperature... 27 1.11 Mohr s Equation.... 28 1.12 Duality... 29 1.13 Principle of Virtual Forces Versus Betti... 30 1.14 Weak and Strong Influence Functions... 32 ix

x Contents 1.15 The Canonical Boundary Values... 37 1.16 The Dimension of the f i... 40 1.17 Reduction of the Dimension of a Problem... 40 1.18 Boundary Element Method... 42 1.19 Finite Elements and Boundary Elements.... 45 1.20 Must Virtual Displacements Be Small?.... 47 1.21 Only When in Equilibrium?... 48 1.22 What is a Displacement and What is a Force?... 49 1.23 The Number of Force and Displacement Terms... 50 1.24 Why the Minus in EA u 00 ¼ p?.... 50 1.25 The Virtual Interior Energy... 51 1.26 Equilibrium.... 52 1.27 How a Mathematician Discovers the Equilibrium Conditions... 54 1.28 The Mathematics Behind the Equilibrium Conditions.... 55 1.29 Sinks and Sources.... 56 1.30 The Principle of Minimum Potential Energy... 56 1.30.1 Minimum or Maximum?... 58 1.30.2 Cracks... 60 1.30.3 The Size of the Trial Space V... 62 1.31 Infinite Energy... 65 1.32 Sobolev s Embedding Theorem... 69 1.33 Reduction Principle... 72 1.34 Nonlinear Problems or Symmetry Lost.... 74 References.... 75 2 Betti s Theorem... 77 2.1 Basics... 77 2.2 Influence Functions for Displacement Terms... 80 2.2.1 Derivation of an Influence Function... 81 2.3 Influence Functions for Force Terms... 84 2.3.1 Influence Function for NðxÞ.... 86 2.3.2 Influence Function for MðxÞ... 88 2.3.3 Influence Functions for Higher-Order Derivatives... 89 2.3.4 Moments Differentiate Influence Functions... 89 2.4 Statically Determinate Structures... 92 2.4.1 Pole-Plans... 95 2.4.2 Construction of Pole-Plans and the Shape of the Displaced Figure... 96 2.4.3 How to Determine the Magnitude of Rotations... 96 2.4.4 Influence Function for a Shear Force (Fig. 2.20).... 99 2.4.5 Influence Function for a Normal Force (Fig. 2.21)... 100 2.4.6 Influence Function for a Moment (Fig. 2.22)... 101 2.4.7 Influence Function for a Moment (Fig. 2.23)... 103 2.4.8 Influence Function for a Shear Force (Fig. 2.24).... 103

Contents xi 2.4.9 Influence Function for Two Support Reactions (Fig. 2.25)... 103 2.4.10 Abutment Reaction (Fig. 2.26)... 106 2.5 Statically Indeterminate Structures... 107 2.6 Influence Functions for Support Reactions... 109 2.7 The Zeros of the Shear Force... 112 2.8 Dirac Deltas... 113 2.9 Dirac Energy... 115 2.10 Point Values in 2-D and 3-D... 122 2.11 Duality... 123 2.12 Monopoles and Dipoles... 125 2.13 Influence Functions for Integral Values... 131 2.14 Influence Functions Integrate... 133 2.15 Second-Order Beam Theory... 135 References.... 137 3 Finite Elements... 139 3.1 The Idea of the FE-Method... 139 3.2 Why the Nodal Values Are Exact.... 142 3.3 Adding the Local Solution... 145 3.4 Projection... 148 3.5 Equivalent Nodal Forces.... 150 3.6 Fixed-End Forces... 152 3.7 Shape Forces and the FE-Load Case... 153 3.8 Slabs and the FE-Load Case.... 158 3.9 Computing Influence Functions with Finite Elements.... 160 3.10 Functionals... 162 3.11 Weak and Strong Influence Functions... 165 3.12 Examples... 165 3.13 The Local Solution... 173 3.14 The Central Equation... 175 3.15 State Vectors and Measurements... 177 3.16 Maxwell s Theorem... 179 3.17 The Inverse Stiffness Matrix... 182 3.18 Examples... 182 3.19 General Form of an FE-Influence Function.... 186 3.20 Finite Differences and Green s Functions... 187 3.21 Stresses Jump, Displacements Don t.... 189 3.22 The Path from the Source Point to the Load... 190 3.23 The Inverse Stiffness Matrix as an Analysis Tool.... 193 3.24 Mohr and the Flexibility Matrix F ¼ K 1... 196 3.25 Non-uniform Plates.... 198 3.26 Sensitivity Plots... 200

xii Contents 3.27 Support Reactions.... 201 3.28 Influence Function for a Rigid Support.... 204 3.29 Influence Function for an Elastic Support.... 208 3.30 Accuracy of Support Reactions.... 211 3.31 Point Loads and Point Supports in Plates... 211 3.32 Point Supports are Hot Spots... 215 3.33 The Amputated Dipole... 216 3.34 Single Forces as Nodal Forces... 221 3.35 The Limits of FE-Influence Functions... 222 References.... 222 4 Betti Extended... 223 4.1 Proof.... 224 4.2 At Which Points is the FE-Solution Exact?... 226 4.3 Exact Values.... 231 4.4 Error at the Nodes... 231 4.5 One-Dimensional Problems.... 233 4.6 Plates and Slabs... 235 4.7 Point Supports of Plates and Slabs... 237 4.8 If the Solution Lies in V h... 239 4.9 Adaptive Refinement... 242 5 Stiffness Changes and Reanalysis... 245 5.1 A First Try... 245 5.2 Second Example... 247 5.3 Strategy... 248 5.4 Variations in the Stiffness... 249 5.5 Dipoles and Monopoles... 250 5.6 Displacement Terms and Force Terms... 252 5.7 The Decay of the Effects... 253 5.8 The Relevance of These Results.... 254 5.9 Frames... 258 5.10 Rigid Support... 258 5.11 The Force Method... 261 5.12 Replacement as Alternative.... 264 5.13 Engineering Approach... 265 5.14 Local Analysis... 267 5.15 Observables... 270 5.16 Springs... 272 5.17 How a Weak Influence Function Operates... 273 5.18 Close by and Far Away... 273 5.19 Summary... 275

Contents xiii 5.19.1 Loss of a Hinged Support... 275 5.19.2 Loss of a Clamped Support... 275 5.19.3 Change in a Spring... 276 5.19.4 Change of a Torsional Spring... 276 5.19.5 Change in the Longitudinal Stiffness of a Bar... 276 5.19.6 Change in the Bending Stiffness of a Beam... 276 5.19.7 Calculating the Deflection w c of a Spring... 277 5.20 Optimal Shape of a Structural Member.... 279 5.21 One-Click Reanalysis... 282 5.21.1 Modifications on the Diagonal.... 282 5.21.2 Plastic Hinges... 283 References.... 284 6 Singularites... 285 6.1 Singular Stresses.... 285 6.2 A Paradox?.... 286 6.3 Single Forces... 286 6.4 The Decay of the Stresses... 290 6.5 Cantilever Beam... 294 6.6 Infinitely Large Stresses... 294 6.7 Symmetry of Adjoint Effects... 299 6.8 Cantilever Plate... 299 6.9 Standard Situations... 303 6.10 Singularities in Influence Functions... 304 Reference... 310 7 Energy Principles of Plates and Slabs and Supplements... 311 7.1 Sign Conventions... 313 7.2 Basic Operations.... 314 7.3 Gateaux Derivative... 316 7.4 Potential Energy... 316 7.5 Galerkin... 317 7.6 Timoshenko Beam... 319 7.7 Laplace Operator... 320 7.8 Linear Elasticity... 321 7.9 Kirchhoff Plate... 323 7.10 Reissner Mindlin Plate... 325 7.11 Geometrically Nonlinear Beam.... 326 7.12 Geometrically Nonlinear Kirchhoff Plate... 328 7.13 Nonlinear Theory of Elasticity... 329 7.14 Green s First Identity and Finite Elements... 331 7.14.1 Potential Energy... 332 7.14.2 Galerkin.... 332 7.14.3 Stiffness Matrices... 333

xiv Contents 7.15 Supplements... 333 7.15.1 Single Force Acting on a Plate... 334 7.15.2 Multipoles... 335 Reference... 336 8 Postscript... 337 Reference... 340 Erratum to: Statics and Influence Functions From a Modern Perspective.... E1 Bibliography... 341 Index... 343